NonExpanding

Nonexpanding is an adjective used to describe something that does not expand or increase in size, extent, or distance. In mathematics and related fields, the term is most often applied to maps between metric or normed spaces, where it is synonymous with a 1-Lipschitz map.

Formally, a function f between metric spaces (X, d_X) and (Y, d_Y) is nonexpanding if for all x1, x2 in X, d_Y(f(x1), f(x2)) ≤ d_X(x1, x2). This means the function does not stretch distances. A related concept is a contraction, which satisfies d_Y(f(x1), f(x2)) < c d_X(x1, x2) for some constant c < 1; every contraction is nonexpanding, but not every nonexpanding map is a contraction.

Nonexpanding maps arise in various contexts. The identity map on any metric space is nonexpanding, and any isometry is nonexpanding since it preserves distances. The composition of nonexpanding maps is nonexpanding. In Hilbert and Banach spaces, specific nonexpansive constructions are important: for example, the metric projection onto a closed convex set in a Hilbert space is nonexpansive, and in stronger terms, firmly nonexpansive.

In fixed-point theory, nonexpansive maps are central to several results. While the Banach fixed-point theorem applies to contractions, many nonexpanding mappings still have fixed points under appropriate conditions, such as in closed convex subsets of uniformly convex spaces, with results attributed to Browder, Göhde, and Kirk among others.

The term nonexpanding is sometimes used interchangeably with nonexpansive, though some authors reserve nonexpansive for the standard term. It is commonly encountered in analysis, metric geometry, and optimization literature.